Topic #1: Evaluating and Simplifying Algebraic Expressions

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1 John Jay College of Criminal Justice The City University of New York Department of Mathematics and Computer Science MAT College Algebra Departmental Final Examination Review Topic #1: Evaluating and Simplifying Algebraic Expressions Evaluate the algebraic expression for the given value or values of the variable(s). y - 7x 1) ; x = - and y = 3 1) 6x + xy A) B) 11 6 C) - 1 D) ) -b + b - 4ac a when a = 5, b = 14, and c = -3 ) A) 1 5 B) C) -3 D) 3 Simplify the algebraic expressions: 3) (1y + 9) + (11y - 6y + 9) 3) A) 11y + 18y - 18 B) 9y 6 C) 11y - 6y + 18 D) 11y + 6y+ 18 4) (3a - b - 5c) - (9a - 6b - 7c) 4) A) -6a + 4b + c B) 1a - 8b - 1c C) -6a + 4b - 1c D) -6a - 8b + c 5) (x - 11)(x + 7x - 5) 5) A) x 3-4x - 8x + 55 B) x x + 8x + 55 C) x x + 7x - 55 D) x 3-4x - 7x ) -35x + 8x ) A) -5x + 4x + 3 B) -35x + 8x + 3 C) -45x + 196x D) -5x + 8x + 1 7) 0x9 y 11 z 9 4x 4 y 3 z 8 7) A) 5x 4 y 7 z B) 5x 5 y 8 C) 5x 5 y 8 z D) x 5 y 8 z 1

2 8) 5x 13 y 6 5x 3 y 3 0 8) A) x 10 y 3 B) 5x 10 y 3 C) 1 D) 0 Simplify the exponential expressions: Topic #: Integer Exponents 9) (-6x4)(8x7) 9) A) -48x 8 B) -48x 11 C) 48x 11 D) 48x 8 10) (-5x 5 y -6 )(x -1 y) 10) A) -10x6 y 7 B) -3x4 y 5 C) -10x4 y 5 D) -10x 4 y 7 11) 1x13 y 13 7x 1 y ) A) 3xy 3 B) 3x 5 y 3 C) 3xy 3 D) 1xy 3 Topic #3: Rational Exponents and Radicals Evaluate the expression : 1) ) A) 13 B) 169 C) 17 D) 119 Add or subtract terms whenever possible. 13) ) A) 10 B) 30 C) -30 D) 0 14) x + 6 8x - 3x 14) A) 5 4x B) 4 4x C) 5 x D) 4 x Rationalize the denominator. 3 15) 7-15) A) B) C) D) Simplify the radical expression. 16) 3 x 8 16) A) x 3 x B) x 3 x C) x 3 x D) x 3 x

3 Evaluate the expressions : 17) 161/4 17) A) 8 B) 16 C) 3 D) 18) 49-3/ 18) 1 A) B) -343 C) 343 D) Simplify by reducing the index of the radical. 19) 0 x 16 19) A) 4 x 4 B) 5 x C) 5 x 4 D) 4 x 0) 8 16x 4 0) A) x B) 1 4x C) 4 x D) x Topic #4: Factoring Factor out the greatest common factor. 1) 1x 4-6x x 1) A) 3(7x 4 - x 3 + 5x ) B) x (1x - 6x + 15) C) 3x(7x 3 - x + 5x) D) 3x (7x - x + 5) Factor by grouping. ) x 3 + 9x - 3x - 7 ) A) (x - 3)(x + 9) B) (x - 3)(x + 9) C) (x - 3)(x - 9) D) (x + 3)(x + 9) Factor the trinomial, or state that the trinomial is prime. 3) x - 1x + 7 3) A) (x + 9)(x - 3) B) (x + 9)(x + 1) C) (x - 9)(x - 3) D) prime 4) 6x + 13x + 6 4) A) (6x + )(x + 3) B) (3x - )(x - 3) C) (3x + )(x + 3) D) prime Factor the difference of two squares. 5) 49x - 16y 5) A) (7x + 4y) B) (7x + 4y)(7x - 4y) C) (7x - 4y) D) prime 3

4 Factor using the formula for the sum or difference of two cubes. 6) 64x 3-1 6) A) (4x + 1)(16x - 4x + 1) B) (4x - 1)(16x + 4x + 1) C) (4x - 1)(16x + 1) D) prime 7) 15x ) A) (5x - 1)(5x + 1) B) (5x + 1)(5x - 5x + 1) C) (5x - 1)(5x + 5x + 1) D) prime Topic #5: Rational Expressions Perform the indicated operations and simplify the result. Leave the answer in factored form. 8) 4x - 4 8x x 5x - 5 A) 0x + 40x + 0 8x 3 B) 3x3-3x 5x - 5x C) 5 3x D) 3x 5 8) 9) x - 10x + 4 x - 1x x - 14x + 4 x - 16x ) A) (x - 4)(x - ) (x - 9)(x - 10) C) (x - 10x + 4)(x - 14x + 4) (x - 1x + 108)(x - 16x + 60) B) D) (x + 4)(x + ) (x + 9)(x + 10) (x - 4) (x - 10) Add or subtract as indicated ) x + - 3x + x - 1 A) C) 6x - 9 (x - 1)(x + 1)(x - ) 9x - 6 (x - 1)(x + 1)(x - ) B) D) 9x - 6 (x - 1)(x - ) 40x - 6 (x - 1)(x + 1)(x - ) 30) 31) x x x + 5x + 4 A) C) x - 4 (x - 4)(x + 4)(x + 1) x - 4x + 0 (x - 4)(x + 4)(x + 1) B) x + 4x + 0 (x - 4)(x + 4)(x + 1) D) x - 4x + 0 (x - 4)(x + 4) 31) 4

5 Topic #6: Complex Numbers Add or subtract as indicated and write the result in standard form. 3) -7 - (- - 7i) - (- + 5i) 3) A) 4 - i B) -3 - i C) -3 + i D) 4 + i Find the product and write the result in standard form. 33) (-3-7i)( + i) 33) A) 1-17i B) i C) i D) i Divide and express the result in standard form. 8 34) 4 + i A) i B) i C) i D) i 34) 35) 6-6i 8 + i A) i B) i C) i D) i 35) Perform the indicated operations and write the result in standard form. 36) ) A) 36i B) -13i C) -13 D) 13i 37) A) -1 - i B) 1 + i 6 C) -1 + i 6 D) -1 - i 6 37) Topic #7: Linear, Rational, Radical, Absolute Value, and Literal Equations Solve and check the linear equations. 38) (-5x + 4) - 5 = -4(x - 7) 38) A) {19} B) {- 9} C) {- 6} D) {9} 39) x 5 = x ) A) {-75} B) {150} C) {75} D) {-150} First, write the value(s) that make the denominator(s) zero. Then solve the equation. 40) 10 x = ) x A) x 0, ; 5 6 B) x 0; C) No restrictions; {} D) x 0; {4} 1 4 5

6 Solve the absolute value equation or indicate that the equation has no solution. 41) 3 x - 3 = 18 41) A) {3, -9} B) {9, -3} C) {3} D) Solve the radical equation, and check all proposed solutions. 4) 6x + 55 = x 4) A) - 11 B) C) {11} D) {-5, 11} Solve the formula for the specified variable. 43) F = 9 C + 3 for C 43) 5 A) C = 5 F - 3 B) C = F C) C = 9 5 (F - 3) D) C = 5 (F - 3) 9 44) A = 1 bh for b 44) A) b = h A B) b = A h C) b = A h D) b = Ah 45) S = rh + r for h 45) A) h = S r - 1 B) h = S - r C) h = (S - r) D) h = S - r r 46) P = L + W for W 46) A) W = P - L B) W = P - L C) W = P - L D) W = d - L 6

7 Topic #8: Linear, Compound, and Absolute Value Inequalities Solve the linear inequality. Other than, use interval notation to express the solution set and graph the solution set on a number line. 47) 7x - 6 6x - 47) A) [4, ) B) (-8, ) C) (-, 4] D) (-, 4) 48) -8x + 4 -(3x + 1) 48) A) [3, ) B) (-, 3) C) (3, ) D) (-, 3] 7

8 Solve the compound inequality. Other than, use interval notation to express the solution set and graph the solution set on a number line. 49) 17 5x ) A) (4, 5) B) (-5, -4) C) [4, 5] D) [-5, -4] Solve the absolute value inequality. Other than, use interval notation to express the solution set and graph the solution set on a number line. 50) x ) A) [-7, 11] B) (-7, 3) C) [-7, 3] D) (-, -7] [3, ) 8

9 51) 7x > -6 51) A) (-, ) B) 6 7, 1 7 C) 6 7, D) 5) 5x < -14 5) A) 3 5, 13 5 B) -, 3 5 C) -, 13 5 D) Topic #9: Distance and Midpoint Formulas; Circles Find the distance between the pair of points. 53) (-1, 4) and (-5, 7) 53) A) 6 B) 5 C) 10 D) 5 9

10 Find the midpoint of the line segment whose end points are given. 54) (7, 3) and (4, 1) 54) A) (11, 4) B) (3, ) C) ( 11, ) D) ( 3, 1) Write the standard form of the equation of the circle with the given center and radius. 55) (-4, 4); 3 55) A) (x - 4) + (y + 4) = 9 B) (x + 4) + (y - 4) = 9 C) (x + 4) + (y - 4) = 3 D) (x - 4) + (y + 4) = 3 Find the center and the radius of the circle. 56) (x - 5) + (y + 7) = 36 56) A) (-5, 7), r = 36 B) (7, -5), r = 36 C) (-7, 5), r = 6 D) (5, -7), r = 6 Complete the square and write the equation in standard form. Then give the center and radius of the circle. 57) x - 1x y - 8y + 16 = 16 57) A) (x - 6) + (y - 4) = 16 (6, 4), r = 4 C) (x - 4) + (y - 6) = 16 (-4, -6), r = 16 B) (x - 6) + (y - 4) = 16 (-6, -4), r = 16 D) (x - 4) + (y - 6) = 16 (4, 6), r = 4 Graph the equation. 58) (x - 1) + (y - ) = 49 58) A) B) Domain = (-6, 8), Range = (-5, 9) Domain = (-8, 6), Range = (-9, 5) 10

11 Topic #10: Basics of Functions and Their Graphs Determine whether the relation is a function. 59) {(-7, -1), (-7, ), (-1, 8), (3, 3), (10, -7)} 59) A) Not a function B) Function Evaluate the function at the given value of the independent variable and simplify. 60) f(x) = -3x - 8; f(-) 60) A) B) - C) 14 D) ) f(x) = x + 11; f(-) 61) A) -3 B) 3 C) 1.73 D) not a real number Use the vertical line test to determine whether or not the graph is a graph in which y is a function of x. 6) 6) A) function B) not a function 63) 63) A) not a function B) function 11

12 64) 64) A) function B) not a function Use the graph to find the indicated function value. 65) y = f(x). Find f(-1) 65) A) -0. B) -4. C) 4. D) 0. 1

13 Use the graph to determine the function's domain and range. 66) 66) A) domain: (-, ) range: [-4, ) C) domain: [-1, ) range: [-4, ) B) domain: (-, ) range: (-, ) D) domain: (-, -1) or (-1, ) range: (-, -4) or (-4, ) 67) 67) A) domain: [0, ) range: [0, ) C) domain: [0, ) range: [-1, ) B) domain: [0, ) range: (-, ) D) domain: (-, ) range: [-1, ) 13

14 Identify the intervals where the function is changing as requested. 68) Increasing 68) A) (-3, 3) B) (-3, ) C) (-, ) D) (-, ) 69) Constant 69) A) (-1, 1) B) (, ) C) (-, -1) D) (1, ) Evaluate the piecewise function at the given value of the independent variable. 70) f(x) = 3x + 1 if x < -1 ; f() 70) -x - 5 if x -1 A) -8 B) -9 C) 1 D) -3 Determine whether the given function is even, odd, or neither. 71) f(x) = x3-5x 71) A) Neither B) Even C) Odd 7) f(x) = x + x4 7) A) Odd B) Neither C) Even 73) f(x) = x3 - x 73) A) Neither B) Odd C) Even 14

15 Topic #11: Slope and Linear Functions Find the slope of the line that goes through the given points. 74) (-, -6), (-9, -17) 74) A) 11 7 B) C) 7 11 D) 3 11 Use the given conditions to write an equation for the line in point-slope form. 75) Slope = 4, passing through (-3, 7) 75) A) x - 7 = 4(y + 3) B) y = 4x + 19 C) y + 7 = 4(x - 3) D) y - 7 = 4(x + 3) Use the given conditions to write an equation for the line in slope-intercept form. 76) Slope =, passing through (7, 3) 76) 3 A) y = 3 x + 7 B) y = 3 x C) y = mx D) y = 3 x ) Passing through (-8, -) and (-5, -7) 77) A) y = x - 46 B) y + = - 5 (x + 8) 3 3 C) y = mx D) y = 5 3 x

16 Graph the line whose equation is given. 78) y = x - 78) A) B) C) D) Determine the slope and the y-intercept of the graph of the equation. 79) 7x - 10y - 70 = 0 79) A) m = 10 7 ; (0, 10) B) m = 7; (0, 70) C) m = 7 10 ; (0, -7) D) m = - 7 ; (0, 7) 10 Use the given conditions to write an equation for the line in the indicated form. 80) Passing through (, 3) and parallel to the line whose equation is y = -x + 3 ; point-slope form A) y - = -(x - 3) B) y - 3 = -(x - ) C) y = x D) y - 3 = x - 80) 16

17 81) Passing through (5, 3) and perpendicular to the line whose equation is y = x + 7; point-slope form 81) A) y - 3 = 1 (x + 5) B) y - 3 = - 1 (x - 5) C) y - 5 = 1 (x - 3) D) y = - x - 11 Find the average rate of change of the function from x1 to x. 8) f(x) = -3x - x from x1 = 5 to x = 6 8) A) B) -34 C) - D) 1 Find and simplify the difference quotient of f, f(x + h) - f(x), h 0, for the function. h 83) f(x) = 4x 83) A) 4 B) 4(x + xh + h ) h C) 4(x+h) D) 8 h + x + 4h 17

18 Topic #1: Transformations of Graphs Begin by graphing the standard quadratic function f(x) = x. Then use transformations of this graph to graph the given function. 84) h(x) = (x - 7) ) A) B) C) D) 18

19 Use the graph of the function f, plotted with a solid line, to sketch the graph of the given function g. 85) g(x) = -f(x - 1) + 85) y = f(x) A) B) C) D) Topic #13: Algebra of Functions, Function Composition, and Inverse Functions Given functions f and g, perform the indicated operations. 86) f(x) = 3-5x, g(x) = -8x + 5 Find f + g. A) -5x B) 3x + 8 C) -8x + 3 D) -13x ) For the given functions f and g, find the indicated composition. 87) f(x) = 3x + 9, g(x) = 5x - 1 (f g)(x) A) 15x + 8 B) 15x + 44 C) 15x + 6 D) 15x ) 19

20 88) f(x) = x + x +, g(x) = x - x - 3 (f g)(-3) A) 51 B) 136 C) 170 D) 17 88) The function f is one-to-one. Find its inverse. 89) f(x) = 3x ) A) f-1(x) = x B) f-1(x) = x 3-7 C) f -1(x) = x D) f -1(x) = x ) f(x) = x ) A) f-1(x) = x + 7, x 0 B) f-1(x) = (x + 7) C) f-1(x) = x - 7, x 0 D) f-1(x) = x ) f(x) = 3x - 7 8x + 4 A) f -1 (x) = -4x - 7 8x - 3 C) f -1 (x) = 8x - 3-4x - 7 B) f -1 (x) = 3x - 7 8x + 4 D) f -1 (x) = 3x + 3 8x ) Solve the equation by factoring. Topic #14: Quadratic Equations and Quadratic Functions 9) x = x + 6 9) A) {-, 3} B) {1, 6} C) {-, -3} D) {, 3} Solve the equation by factoring. 93) x + x - 10 = 0 93) A) {1, -10} B) {-1, 1} C) {1, 10} D) {-1, 10} Solve the equation by the square root property. 94) 6x = 54 94) A) {-3 6, 3 6} B) {-6, 6} C) {-3, 3} D) {0} 95) (x - 3) = 49 95) A) {5} B) {-10, -4} C) {-7, 7} D) {-4, 10} Solve the equation by completing the square. 96) x + 14x + 6 = 0 96) A) {-7-3, } B) {7-6, 7 + 6} C) { } D) {7 + 3} 0

21 Solve the equation using the quadratic formula. 97) x + 7x + 7 = 0 97) A) , B) , C) -7-77, D) 7-1, ) 5x - 3x + 3 = 0 98) A) 3 ± i B) -3 ± C) -3 ± i D) 3 ± The graph of a quadratic function is given. Determine the function's equation. 99) 99) A) h(x) = (x - ) + B) g(x) = (x + ) - C) j(x) = (x - ) - D) f(x) = (x + ) + 100) 100) A) f(x) = -x - x - 1 B) g(x) = -x + x + 1 C) j(x) = -x + 1 D) h(x) = -x - 1 Find the coordinates of the vertex for the parabola defined by the given quadratic function. 101) f(x) = (x - 4) ) A) (0, -4) B) (4, 4) C) (4, -4) D) (-4, 0) 1

22 10) y + 4 = (x - ) 10) A) (, - 4) B) (-, - 4) C) (4, - ) D) (4, ) Find the axis of symmetry of the parabola defined by the given quadratic function. 103) f(x) = x ) A) x = -7 B) x = 7 C) x = 0 D) y = 7 104) f(x) = (x + 4) ) A) x = 6 B) x = -6 C) x = -4 D) x = 4 Topic #15: Introduction to Polynomial and Rational Functions Form a polynomial whose zeros and degree are given. 105) Zeros: -3, -, ; degree 3 105) A) f(x) = x 3-3x + 4x - 1 for a = 1 B) f(x) = x 3 + 3x + 4x + 1 for a = 1 C) f(x) = x 3-3x - 4x + 1 for a = 1 D) f(x) = x 3 + 3x - 4x - 1 for a = 1 For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. 106) f(x) = 5(x + 3)(x - 3) 3 106) A) -3, multiplicity 1, crosses x-axis; 3, multiplicity 3, crosses x-axis B) 3, multiplicity 1, touches x-axis; -3, multiplicity 3 C) 3, multiplicity 1, crosses x-axis; -3, multiplicity 3, crosses x-axis D) -3, multiplicity 1, touches x-axis; 3, multiplicity 3 107) f(x) = (x + 4)(x + 1) 107) A) -1, multiplicity, touches x-axis B) -1, multiplicity, crosses x-axis C) -4, multiplicity 1, crosses x-axis; -1, multiplicity, touches x-axis D) -4, multiplicity 1, touches x-axis; -1, multiplicity, crosses x-axis Find the x- and y-intercepts of f. 108) f(x) = (x + 4)(x - )(x + ) 108) A) x-intercepts: -4, -, ; y-intercept: -16 B) x-intercepts: -,, 4; y-intercept: -16 C) x-intercepts: -4, -, ; y-intercept: 16 D) x-intercepts: -,, 4; y-intercept: ) f(x) = 4x - x 3 109) A) x-intercepts: 0, -4; y-intercept: 0 B) x-intercepts: 0,, -; y-intercept: 0 C) x-intercepts: 0,, -; y-intercept: 4 D) x-intercepts: 0, -4; y-intercept: 4 List the potential rational zeros of the polynomial function. Do not find the zeros. 110) f(x) = 6x 4 + x 3-3x + 110) A) ± 1 6, ± 1 3, ± 1, ± 3, ± 1, ±, ± 3 B) ± 1 6, ± 1 3, ± 1, ± 1, ± C) ± 1 6, ± 1 3, ± 1, ± 3, ± 1, ± D) ± 1, ± 3, ± 1, ±, ± 3, ± 6

23 Use the Remainder Theorem to find the remainder when f(x) is divided by x - c. 111) f(x) = x 4 + 8x 3 + 1x ; x ) A) R = 1 B) R = -1 C) R = -5 D) R = 5 Form a polynomial f(x) with real coefficients having the given degree and zeros. 11) Degree 3: zeros: 1 + i and -5 11) A) f(x) = x3 + x - 8x + 10 B) f(x) = x3-5x - 8x - 1 C) f(x) = x3 + 3x - 8x + 10 D) f(x) = x3 + 3x + 10x - 8 Use the given zero to find the remaining zeros of the function. 113) f(x) = x4-1x - 100; zero: -i 113) A) i, 5i, -5i B) i, 10, -10 C) i, 10i, -10i D) i, 5, -5 Divide using synthetic division. 114) x4-3x 3 + x + 4x - 5 x - 1 A) x 3 + x - x C) x 3 - x + x x x - 1 B) x 3 - x + x D) x 3 - x - x x - 1 x ) Use the Leading Coefficient Test to determine the end behavior of the polynomial function. 115) f(x) = 3x 4 + 4x 3-4x + 3x - 115) A) rises to the left and rises to the right B) falls to the left and rises to the right C) falls to the left and falls to the right D) rises to the left and falls to the right 116) f(x) = x 3 + 5x + 5x ) A) falls to the left and rises to the right B) falls to the left and falls to the right C) rises to the left and falls to the right D) rises to the left and rises to the right Find the domain of the rational function. 117) g(x) = x x + A) {x x -} B) all real numbers C) {x x } D) {x x 0} 117) 118) f(x) = x + 7 x - 9 A) {x x -3, x 3, x -7} B) all real numbers C) {x x -3, x 3} D) {x x 0, x 9} 118) 119) f(x) = x + x + 16x A) {x x 0, x -16} B) {x x -4, x 4, x -} C) {x x -4, x 4} D) all real numbers 119) 3

24 Find the vertical asymptotes of the rational function. 4x 10) h(x) = (x + )(x - 6) A) x = -, x = 6 B) x = -, x = 6, x = -4 C) x =, x = -6 D) x = -4 10) 11) g(x) = x + 4 x + 4 A) x = -, x =, x = -4 B) none C) x = -, x = -4 D) x = -, x = 11) 4

25 Answer Key Testname: MAT105_FINAL EXAM_REVIEW 11 1) A ) A 3) D 4) A 5) A 6) A 7) C 8) C 9) B 10) C 11) C 1) C 13) B 14) C 15) D 16) C 17) D 18) A 19) C 0) D 1) D ) A 3) C 4) C 5) B 6) B 7) B 8) D 9) A 30) C 31) C 3) C 33) A 34) B 35) D 36) D 37) D 38) B 39) C 40) B 41) B 4) C 43) D 44) B 45) D 46) B 47) A 48) A 5

26 Answer Key Testname: MAT105_FINAL EXAM_REVIEW 11 49) C 50) C 51) A 5) D 53) D 54) C 55) B 56) D 57) A 58) A 59) A 60) B 61) B 6) B 63) A 64) A 65) C 66) A 67) C 68) D 69) A 70) B 71) C 7) C 73) A 74) A 75) D 76) B 77) A 78) D 79) C 80) B 81) B 8) B 83) C 84) B 85) D 86) D 87) C 88) C 89) A 90) C 91) A 9) A 93) D 94) C 95) D 96) A 6

27 Answer Key Testname: MAT105_FINAL EXAM_REVIEW 11 97) B 98) A 99) D 100) C 101) C 10) A 103) C 104) C 105) D 106) A 107) A 108) A 109) B 110) C 111) D 11) C 113) D 114) D 115) A 116) A 117) A 118) C 119) A 10) A 11) B 7